In this section, we follow the approach presented in Ref. [1]} to obtain the lattice distortion when the twist angle is near the first magic angle. We assume that the lattice distortion is independent of the sublattice labeled by \(S=A\)  or \(S=B\) , i.e. \(U^{\parallel ,\perp }_{j, S}(x) = U^{\parallel ,\perp }_j(x)\) , where \(j=t\)  refers to the top layer and \(j=b\)  refers to the bottom layer. We further neglect the lattice corrugation, so that \(U^{\perp }_b = 0\)  and \(U^{\perp }_t = d_0 \hat{z}\) . Under these assumptions, the intra-layer elastic energy of the graphene system can be written as
\(U_E = & \frac{1}{2}\sum _{j = t, b} \int {\rm d}^2 x \left[ \mathcal {K} (\partial _x U^{\parallel }_{j, x} + \partial _y U^{\parallel }_{j, y})^2 + \right. \nonumber \\& \left. \mathcal {G} \left( (\partial _x U^{\parallel }_{j, x} - \partial _y U^{\parallel }_{j, y})^2 + (\partial _y U^{\parallel }_{j, x} + \partial _x U^{\parallel }_{j, y})^2 \right) \right]\) 
